Fermionic Symmetry Protected Topological Phases and Cobordisms Anton Kapustin∗ Ryan Thorngren† Alex Turzillo‡ Zitao Wang§ August 19, 2015 Abstract It has been proposed recently that interacting Symmetry Protected Topological Phases can be classified using cobordism theory. We test this proposal in the case of Fermionic SPT phases with Z2 symmetry, where Z2 is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known Fermionic SPT phases in space dimension D ≤ 3 and also predicts that all such phases can be realized by free fermions. In higher dimensions we predict the existence of inherently interacting fermionic SPT phases. 1 Introduction Classification of Symmetry Protected Topological Phases has been a subject of intensive activity over the last few years. In the case of arXiv:1406.7329v2 [cond-mat.str-el] 18 Aug 2015 free fermions, a complete classification has been achieved in [1, 2] using such ideas as Anderson localization and K-theory. In the case of bosonic systems, all SPT phases are intrinsically interacting, so one ∗California Institute of Technology, Pasadena, CA †University of California, Berkeley, CA ‡California Institute of Technology, Pasadena, CA §California Institute of Technology, Pasadena, CA 1 has to use entirely different methods. Interactions are also known to affect fermionic SPT phases [3, 8, 4, 5]. Recently it has been proposed that cobordism theory can provide a complete classification of both bosonic and fermionic interacting SPT phases in all dimensions. This improves on the previous proposal that group cohomology classifies interacting bosonic SPT phases [6], while “group supercohomology” [7] classifies interacting fermionic SPT phases. For bosonic systems with time-reversal and U(1) symmetries the cobordism proposal has been tested in [9] and [10] respectively. Cobordism theory has been found to describe all known bosonic SPT phases with such symmetries in D ≤ 3. In this paper we test the proposal further by studying fermionic SPT phases with Z2 symmetry. The Z2 symmetry in question can be either unitary or anti-unitary. In the former case we will assume that the symmetry is internal (does not act on space-time). In the latter case it must reverse the direc- tion of time, so we will call it time-reversal symmetry. In either case, the generator can square either to 1 or to (−1)F (fermion parity). Fermionic SPT phases with time-reversal symmetry are also known as topological superconductors, so in particular we describe a classifica- tion scheme for interacting topological superconductors. Compared to the bosonic case, fermionic SPT phases present sev- eral related difficulties. First of all, one needs to decide what one means by a fermionic system. In a continuum Lorentz-invariant field theory, anti-commuting fields are also spinors with respect to the Lorentz group, but condensed matter systems are usually defined on a lattice and lack Lorentz invariance on the microscopic level. Thus the connection between spin and statistics need not hold. A related issue is that all fermionic systems have Z2 symmetry called fermionic parity, usually denoted (−1)F . But all observables, including the Hamilto- nian and the action, are bosonic, i.e. invariant under (−1)F . Ina sense, every fermionic system has a Z2 gauge symmetry, which means that the partition function must depend on a choice of a background Z2 gauge field. It is tempting to identify this gauge field with the spin structure. However, it is not clear how a spin structure should be defined for a lattice system, except in the case of toroidal geometry.1 Instead of dealing with all these difficult questions, in this paper we take a more “phenomenological” approach: we make a few assump- 1In 2d, there is a good combinatorial description of spin structures via so called Kaste- leyn orientations [11]. But a generalization of this construction to higher dimensions is unknown. 2 tions about the long-distance behavior of SPT phases which parallel those for bosonic SPT phases, and then test these assumptions by com- paring the results in space-time dimensions d ≤ 4 with those available in the condensed matter literature. For various reasons, we limit our selves to the cases of no symmetry, time-reversal symmetry, and uni- tary Z2 symmetry. Having found agreement with the known results, we make a conjecture about the classification of fermionic SPT phases with any symmetry group G. R. T. would like to thank Xie Chen for patiently answering his questions and Rob Kirby for an enlightening discussion. A. K. would like to acknowledge conversations with Alexei Kitaev and Zhengcheng Gu. The work of A. K., A. T., and Z. W. was supported in part DOE grant DE-FG02-92ER40701 2 Spin and Pin structures A smooth oriented d-manifold M equipped with a Riemannian met- ric is said to have a spin structure if the transition functions for the tangent bundle, which take values in SO(d), can be lifted to Spin(d) while preserving the cocycle condition on triple overlaps of coordi- nate charts. Let us unpack this definition. On a general manifold one cannot choose a global coordinate system, so one covers M with coordinate charts Ui, i ∈ I. If over every coordinate chart Ui one picks an orthonormal basis of vector fields with the correct orienta- tion, then on double overlaps Uij = Ui Uj they are related by tran- sition functions gij which take values inT the group SO(d) and satisfy on Uijk = Ui Uj Uk the cocycle condition: T T gijgjk = gik. (1) The group SO(d) has a double cover Spin(d), i.e. one has SO(d) = Spin(d)/Z2. One can lift every smooth function gij : Uij → SO(d) to a smooth function hij : Uij → Spin(d), with a sign ambiguity. Thus on every Uijk one has hijhjk = ±hik. (2) M has a spin structure if and only if one can choose the functions hij so that the sign on the right-hand side is +1 for all Uijk. We also identify spin structures which are related by Spin(d) gauge transformations: ′ −1 hij 7→ hij = hihijhj , hi : Ui → Spin(d). 3 A spin structure allows one to define Weyl spinors on M. For d < 4 every oriented d-manifold admits a spin structure, but it is not unique, in general. Namely, given any spin structure, one can modify it by multiplying every hij by constants ζij = ±1 satisfying ζijζjk = ζik. Such constants define a Cech 1-cochain on M with values in Z2. The same data also parameterize Z2 gauge fields on M, thus any two spin structures differ by a Z2 gauge field. It is easy to see that gauge fields differing by Z2 gauge transformations lead to equivalent transforma- tions of spin structures, so the number of inequivalent spin structures 1 is equal to the order of the Cech cohomology group H (M, Z2), whose elements label gauge-equivalence classes of Z2 gauge fields. In dimension d > 3 not every oriented manifold admits a spin structure. For example, the complex projective plane CP2 does not admit a spin structure. Nevertheless, if a spin structure on M exists, the above argument still shows that the number of inequivalent spin 1 structures is given by |H (M, Z2)|. The necessary and sufficient con- dition for the existence of a spin structure is the vanishing of the 2nd 2 Stiefel-Whitney class w2(M) ∈ H (X, Z2). This condition is purely topological and thus does not depend on the choice of Riemannian metric on M. If M is not oriented, the transition functions gij take values in O(d) rather than SO(d). They still satisfy (1). An analog of Spin group in this case is called a P in group. In the absence of orientation, fermions transform in a representation of the P in group. In fact, for all d > 0 there exist two versions of the P in group called P in+(d) − ± and P in (d). They both have the property P in (d)/Z2 = O(d). The difference between P in+ and P in− is the way a reflection of any one of coordinate axis is realized on fermions. Let r ∈ O(d) be such a reflection. It satisfies r2 = 1. Ifr ˜ ∈ P in±(d) is a pre-image of r, it can satisfy eitherr ˜2 = 1 orr ˜2 = −1. The first possibility corresponds to P in+, while the second one corresponds to P in−. If we are given an unoriented d-manifold M, we can ask whether it admits P in+ or P in− structures (that is, lifts of transition functions to either P in+(d) or P in−(d) so that the condition (2) on triple overlaps is satisfied). The conditions for this are again topological: in the case + − of P in it is the vanishing of w2(M), while in the case of P in it 2 is the vanishing of w2(M)+ w1(M) . Note that if M happens to be 4 orientable, then w1(M) = 0, so the two conditions coincide and reduce to the condition that M admit a Spin structure. Note that these topological conditions are nontrivial already for d = 2. More precisely, for d = 2 one has a relation between Stiefel- 2 − Whitney classes w1 + w2 = 0, so every 2d manifold admits a P in structure, but not necessarily a P in+ structure. For example the real projective plane RP2 admits only P in− structures, while the Klein bottle admits both P in+ and P in− structures. Similarly, not every 3-manifold admits a P in+ structure, but all 3-manifolds admit a P in− structure. 3 Working assumptions We assume that fermionic SPTs in d space-time dimensions without time-reversal symmetry can be defined on any oriented smooth d- manifold M equipped with a spin structure.